Find the centers of the studs in the bays that they cut the holes in, cut out a square piece of sheetrock the same thickness as the existing, cut a square from center of stud to center of the next stud, secure the new sheetrock with 1-1/2~1-5/8" sheetrock screws, sand the surrounding existing sheetrock back to almost the sheetrock's outer "paper" about 8~12" back from your cut lines.

Once you have it sanded down and even with your patch, put a coat of drywall mud on the seams, then tape the seams and mud over your tape. Don't allow the first coat of mud to dry before taping and re-mudding. This will help give an even finish to skim over.

Once it's taped and mudded, you may need to sand the tape lines depending on the thickness / type of texture.

Put an even skim coat over the patched area and surrounding area and feather it into your existing texture.

Once you are happy with the base, texture it to match the existing, whether it's orange peel or hand texture, put a coat of drywall primer on the areas with new mud and paint.

I prefer to go back to studs instead of just patching the round hole because it allows for the patch to be completely secured and most of the time prevents cracking at your seams, and is also easier to tape squares evenly.

Just my two cents.